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Here is a partial answer. It proves a fault-free rectangle can be assembled from rectangles of size mxn such that one dimension is not a multiple of the other. The remaining cases can be converted to the 1xn case solved earlier by Bubbler.


Using Jaap's interpretation of the problem, I believe First claim: Proof by example: Second claim: Informal proof (might have a flaw): A lemma is needed at this point: Proof: Back to the second claim: As a demonstration of the construction, here are 2×5 tiles covering a 30×22 rectangle and 3×4 tiles covering a 36×27 rectangle, "seamlessly". ...


I left the empty square at A1. I found 25000 tilings that work for A1, out of an estimated 500 000 give or take a few hundred thousand. The rectangles are marked with lighter colours.

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